Chapter IV: Tidal Splicing for BHNS and BNS Systems
4.5 Results
PNR(v) =φNR(v). (4.59) Similar to TaylorT4, we compute the analytic TaylorT2 tidal effects,TTid(v) according to Eq 4.68 andPTid(v) according to Eq 4.69, incorporating the dynamical tides by making the frequency de- pendent adjustment to ¯λ`from Eq 4.28.
The spliced waveform’s timetSpland phaseφSplare then given by examining Eq 4.13 and making the appropriate substitutions,
tSpl=t0+tNR(v)+TTid(v),
φSpl=φ0+φNR(v)+PTid(v). (4.60) We use the freedom inherent within choosingt0 andφ0in order to align the spliced waveform at the initial time to the numerical waveform (though this is somewhat unnecessary as computing mis- matches between waveforms marginalizes over these choices). We terminate the spliced wavefroms at late times according to the simulations we test against, the details of which are given in Sec 4.5.
As the waveform nears the merger phase of the evolution, the effect fromTTid(v) might grow larger than that oftNR(v). At that point the change in time values flatten out then decreases with increasing v leading to unphysical behavior in the wavefrom, so we consider the PN approximation to have broken down and end the waveform there.
Waveform Reconstruction
OncetSpl(v) andφSpl(v) are computed, then the final step is reconstructing the spliced waveform.
The expressions for A`mTid(v) in Eq 4.23 are the tidal corrections to the individual modes, with the dynamical tides accounted for by the replacement rule ¯λ2→λ¯2Aκˆ2A(v) from Eq 4.30.
We then arrive at the final formula for the spliced waveform modes, h`mSpl
tSpl(v)
=
A`mNR(v)+A`mTid(v)
ei(ψ`mNR(v)−mφSpl(v)). (4.61) From here, the amplitudes and phases of each mode are interpolated onto a uniform time grid using a cubic splice.
Type q χNS f0(Hz) f1(Hz) Ncyc22
BHNS 1 0 218 578 19.9
BNS 1 0 211 629 20.8
BHNS 1 -.2 217 505 17.0
BHNS 1.5 0 154 537 28.9
BHNS 2 0 156 505 21.0
BHNS 2 -.2 156 485 19.8
Table 4.1: List of parameters for numerical simulations considered
here. In particular, there are 5 BHNS and 1 BNS runs, and two of the BHNS runs have a small anti-aligned spin on the NS while the rest of the runs are nonspinning.
We generated our tidally spliced waveforms for each of these cases using the hybridized surrogate model ‘NRHybSur3dq8’ [154] to compute the underlying BBH signal and making use of the uni- versal relations to obtain the other tidal parameters from ¯λ2; the details are given in Appendix 4.6.
To provide an additional point of comparison, we also test another waveform model, SEOBNRv4T, which is the time domain model SEOBNRv4 [53] augmented with most of the same effects we have used here, including higher order corrections to the static tides in the EOB potential Eq 4.63 [34], strain corrections Eq 4.23 [24], and dynamical tides (without the resonance frequency correction for spinning NSs) Eq 4.28, 4.30 [73]. The implementation of SEOBNRv4T we used here only outputs the (2,2) mode; however our analysis involves systems with mass ratios near unity so the (2,2) mode should dominate the gravitational waveforms. Thus, we anticipate this limitation should will be a minor effect on our results.
Waveforms
We include all modes from the numerical waveforms up through` = 5, while the surrogate and spliced waveforms use all available modes [(2, 0), (2, 1), (2, 2), (3, 2), (3, 0), (3, 1), (3, 3), (4, 2), (4, 3), (4, 4), (5, 5)], and SEOBNRv4T only has the (2,2) mode. (Since the system is spin-aligned, we only needm≥0 modes, symmetry givesm< 0 modes from that).
We choose the beginning of the waveform to be at a time after the initial burst of junk radiation, t=200M. That time also set the starting orbital frequency of the waveform (chosen according half the the time derivative of the phase of the (2,2) mode). To prevent the starting frequencyωInitialfrom being contaminated by residual junk radiation and slight eccentricities in the imperfect BHNS/BNS initial data, we use a quadradic fit of the simulations’ frequency against time over the first 500M to estimate the precise starting frequency. We window the waveform with a Planck-Taper window over that early 500Mregion of the waveform. We label the orbital frequency at the end of this window
asω0; this frequency will serve as the initial frequency considered in our mismatches below.
At late times, we set the upper frequency cutoffωCutoffby the frequency attained by the simulation at its peak power, and window the waveform (again with Planck-Taper) over the times fromω1 = .85ωCutoff toωCutoff. This gives us an inspiraling waveform from orbital frequencyω0 toω1, the values of which are given in Table 4.1 in physical units as f0and f1as measured in Hz. The other waveforms we generated fromωInitialtoωCutoffand windowed in a similar manner. In Table 4.1, we also list the number of cycles in the (2,2) mode,Ncyc22, in the numerical simulation within the listed orbital frequency bounds. While this is not necessarily a large frequency range to be making our comparisons, it is during the late inspiral where we expect the tidal effects to make the strongest contributions to the binary’s evolution.
After we transform all of the waveforms into the frequency domain, we calculate the mismatches with the numerical waveforms. We follow the procedures used in Appendix D of [41] to evaluate overlap function assuming a 2-detector network, each measuring one of the polarizations with a flat noise spectrum, optimized over time and polarization phase shifts between the waveforms. The starting and ending frequency of the windowed waveforms, as given in Table 4.1, bound the fre- quency range for the mismatch computations. The mismatch between each of the models and the numerical simulation is then computed across a uniform distribution of sky locations.
As a quick aside, we attempted to compare against a pair of frequency domain tidal waveform models, SEOBNRv4ROM_NRTidal and IMRPhenomD_NRTidal, but found mismatches which we believed to be artificially large, near that of the BBH waveform. This is likely a consequence of having short numerical waveforms; because all of the time domain waveforms here are windowed in the same way, any systematic contamination introduced by the windows should be near identical across them all but that conamination will not affect the frequency domain wavefroms. Thus, we have excluded the frequency domain waveforms from the results below.
Mismatch Comparison: All Modes
In Fig 4.2, we plot a histogram for the distribution of mismatches against the numerical wavefrom across sky locations in the case of the equal mass, nonspinning BHNS system. We do not normalize the vertical axis since the exact heights of the histograms are dependent on the binning choice; the location of the histogram peaks correspond to how well the model does while the spread measures how dependent the models is on sky location.
We estimate of the numerical simulation error during the inspiral by the mismatch between the simulation at two different resolutions (blue). Given that those mismatches are around∼10−4, that suggests the simulations is well resolved during the inspiral. (Merger/ringdown might be a different story.) The mismatches with the surrogate BBH waveform (black) measures the strength of the tidal effects in the system, showing how poorly the waveforms will be if tidal effects are neglected entirely.
As expected, both of the tidal splicing methods we try here, TaylorT4 (magenta) and TaylorT2 (red),
Figure 4.2: Distribution of mismatches against the inspiral of the BHNS,q=1, nonspinning simu- lation across sky locations.
and the SEOBNRv4T model (cyan), all improve upon the BBH waveform, though to varying de- grees. TaylorT4 splicing shows the least improvement, accounting for only some of the NSBH/BNS tidal effects, while both TaylorT2 and SEOBNRv4T have mismatches about an order of magnitude smaller. The narrowness of the peaks here likely correspond to the fact in equal mass, nonspinning systems the gravitational radiation is dominated almost entirely by the (2,2) mode regardless of which way the system is oriented.
In Fig 4.3, we observe similar patterns across most of the waveforms we considered. In all cases, the numerical error of the inspiral is well below any of the mismatches from the waveforms considered here, and the size of the tidal effects behaves qualitatively as expected (i.e. more extreme mass ratios have smaller tidal effects, spinning NS has larger tidal effects, . . . ). The heirarchy between the TaylorT2 splicing, SEOBNRv4T, and TaylorT4 splicing, is more or less preserved in all 6 cases as well; with the exception of theq=2 cases, TaylorT4 is distinctly worse than SEOBNRv4T.
Comparing the q = 1 nonspinning BHNS and BNS cases (top- and middle-left) shows very lit-
Figure 4.3: Similar to Fig 4.2, except displaying the mismatch histograms for all 6 numerical simu- lations we consider.
tle change in behavior in the mismatches, with the BNS mismatches slightly larger, presumably because the tidal effects are at least twice as large (since two objects being deformed rather than 1). Increasing the BHNS mass ratio fromq = 1 (top-left) to q = 1.5 (middle-right) andq = 2 (bottom-left), while keeping the system nonspinning shows improvement in the TaylorT4 splicing waveform in comparison to the others. The distribution for SEOBNRv4T widens significantly with the increasing mass ratio, likely owing at least in part to the growing significance of modes beyond (2,2), as we will discuss below.
The most significant change to the mismatches arises in the case of the spinning NS (q = 1 top- right; q = 2 bottom-right). In both cases, the mismatches of the waveforms worsen significantly compared to the corresponding nonspinning cases. At best, the effects included here only account for some of the changes the spinning NS has on the evolution of the system and on the gravitational radiation. Either due to the missing spin-tidal terms (see Appendix 4.6), the inaccurate handling of the dynamical tides in the case of anti-aligned NS, or some other unaccounted tidal effect, further work will need to be done in order to properly capture the full behavior of these systems.
Mismatch Comparison: (2,±2) Modes
In order to characterize how much of the disparity between the TaylorT2 spliced and SEOBNRv4T waveforms is due to the inclusion of higher order modes in the former model, we compute the mismatches after restricting the BBH and spliced waveforms to only the (2,±2) modes (see Fig 4.4).
The numerical simulations still utilize all the same modes as before.
In the nonspinningq= 1 BHNS and BNS systems, there is very little change in the mismatches in the spliced waveforms when excluding the other modes. This is almost certainly because a majority of the power in those waveforms is already concentrated in the (2,2) mode so leaving out the other modes is a negligible change to the result. Thus the difference between the waveforms is within the differences in the models themselves. This also holds true for the spinning systems, as it seems that their handling of the spinning-tidal dynamics limits all of the models.
When moving to higher mass ratios for the nonspinning waveforms, the reverse is true namely TaylorT2 and TaylorT4 became worse with TaylorT2 comparable to SEOBNRv4T. The inclusion of higher order modes accounts entirely for the discrepancy between TaylorT2 spliced and SEOB- NRv4T.
Discussion
A curious feature from these results is the rather disparate mismatch values between the TaylorT2 and TaylorT4 methods. That TaylorT4 splicing does so poorly is also odd considering how well TaylorT4 tend to do particularly well for simulating equal mass, nonspinning BBH waveforms. In principle, the only differences between these TaylorT2 spliced and TaylorT4 spliced waveforms should be the how the different PN approximantes handle the truncation error from missing higher order tidal terms. However, the dynamical tides correction do not have a proper power series ex- pansion and thus the differences in the two splicing methods will not fall along neat PN lines; the
Figure 4.4: Similar to Fig 4.3, except that the BBH and tidally spliced waveforms are generated with only the (2,2) mode.
particular way which we incorporated the dynamical tides may yield different behaviors in the dif- ferent splicing schemes. Another possibility is that the missing PN flux and strain coefficients which we set 0 might contribute to the difference, though given we expect them to be negligible contribu- tions to the waveforms, so we deem it less likely those terms are the guilty culprit. More work will need to go into understanding if the differences between the methods is owing to the higher order truncations, the missing flux or strain coefficients, a consequence of how the dynamical tidal effects are handled, or some other factor.
Overall, the TaylorT2 splicing method show improvement (albeit very modest) over utilizing SEOB- NRv4T in all waveforms considered here while TaylorT4 needs further study before being used for BHNS/BNS systems.
4.6 Appendices